Final answer:
To find the area of the region bounded by the cardioid r = 1 - sin θ in polar coordinates, you can use a double integral. Express the area element dA in terms of polar coordinates and determine the limits of integration for θ. Finally, integrate the area element with respect to θ.
Step-by-step explanation:
To find the area of the region bounded by the cardioid r = 1 - sin θ in polar coordinates, we can use a double integral. First, we can express the area element dA in terms of polar coordinates as dA = (1/2)r^2dθ. Next, we need to determine the limits of integration for θ. Since the cardioid is symmetric about the y-axis, we can integrate from 0 to π. Finally, we integrate the area element dA with respect to θ from 0 to π using the limits determined earlier.